Values Of Fractional Order Research Articles

Dynamical effects of hypergraph links in a network of fractional-order complex systems.

In recent times, the fractional-order dynamical networks have gained lots of interest across various scientific communities because it admits some important properties like infinite memory, genetic characteristics, and more degrees of freedom than an integer-order system. Because of these potential applications, the study of the collective behaviors of fractional-order complex networks has been investigated in the literature. In this work, we investigate the influence of higher-order interactions in fractional-order complex systems. We consider both two-body and three-body diffusive interactions. To elucidate the role of higher-order interaction, we show how the network of oscillators is synchronized for different values of fractional-order. The stability of synchronization is studied with a master stability function analysis. Our results show that higher-order interactions among complex networks help the earlier synchronization of networks with a lesser value of first-order coupling strengths in fractional-order complex simplices. Besides that, the fractional-order also shows a notable impact on synchronization of complex simplices. For the lower value of fractional-order, the systems get synchronized earlier, with lesser coupling strengths in both two-body and three-body interactions. To show the generality in the outcome, two neuron models, namely, Hindmarsh-Rose and Morris-Leccar, and a nonlinear Rössler oscillator are considered for our analysis.

COMPUTATIONAL MODEL OF A FRACTIONAL-ORDER CHEMICAL REACTOR SYSTEM AND ITS CONTROL USING ADAPTIVE SLIDING MODE CONTROL

Dynamics of chemical reactor systems are found with highly nonlinear behavior. Computational modeling of a fractional-order chemical reactor system and investigating nonlinear dynamical changes and its control are the main focus of this research work. Chaos theory is a blooming fertile field in recent years, which is used widely to quantify nonlinear behaviors such as quasi-oscillations, bi-stability and bifurcation. The work starts from deriving state-space model of the system with first-order differential equations. There are six equilibrium points and the Jacobian matrix is derived for investigating the stability of the equilibrium points. Eigenvalues of each equilibrium point are calculated. Based on the sign of the real part of the eigenvalues and the existence of imaginary part, we found two equilibrium points behave as saddle spirals and the remaining four equilibrium points are saddle nodes. The stability of the system for different parameter values is investigated and presented. The influence of parameters in the system dynamics is discussed and significant parameter values are highlighted for further study. We considered Caputo’s definition for formulating the fractional-order (FO) model of the system based on the advantages highlighted in various literatures. The stable and unstable regions are portrayed with parameter variations. The results clarified that the analysis can be refined using fractional-order treatment of chaotic systems. We proceeded with our investigation towards obtaining different oscillations, particularly chaotic oscillations. The challenges lie in finding the proper fractional order to handle the system. We showed the bifurcation diagram for a range of fractional-order values and clarified the transitions from periodic oscillations to chaotic behavior and period-doubling bifurcations. The phase portraits are presented to show the limit cycle oscillations for fractional-order 0.95, period-doubling during 0.98, and chaotic oscillations for higher values. We proceeded with our investigation with fractional-order as 0.99. Bifurcation plots for parameter variation are obtained. Chaotic regions, periodic oscillations, period-halving and period-doubling are observed and the influences are discussed. We emphasize the intricate properties which are not addressed during the integer-order treatment of the system and nail the importance of fractional-order treatment. An adaptive sliding mode (ASM) controller is derived and implemented to control the system precisely. The effectiveness is shown by providing simulation results of the system with parameter estimation and controlled state time history plots. The work can be extended to verify the simulated results with equivalent electronic circuits.

Fractional HBV infection model with both cell-to-cell and virus-to-cell transmissions and adaptive immunity

In this work, a fractional order hepatitis B virus infection model with both cell-to-cell and virus-to-cell transmissions and adaptive immunity will be examined. The adaptive immunity that we will consider will be represented by the cellular and the humoral immune responses. In our model, the two modes of infection will be symbolized by two saturated incidence functions. We started our study by proving the existence, uniqueness and boundedness of the positive solutions. Next, we have formulated the free-equilibrium and the endemic equilibria of our model. By using Lyapunov’s method and LaSalle’s invariance principle, we have shown the global stability of each equilibrium. Numerical simulation is also given to support our theoretical results and to show the effect of the fractional derivative order on the convergence toward the equilibrium points. More precisely, numerical results have confirmed our theoretical findings about the equilibria stability. We have noticed that for smaller fractional derivative order values, the variables of our model converge more quickly to their corresponding steady states. However, for a higher values of the fractional derivative order, the convergence becomes very slowly, this means a long memory effect. In other words, the fractional derivative order has no effect on the equilibria stability but only on the convergence speed toward the equilibria. In order to show the importance of incorporating to saturated infection rates, a numerical comparison between the dynamical behavior of the model with two saturated incidence and two bilinear incidence rates is curried out.

SOME CHAOTIC MATHEMATICAL MODELS WITH STOCHASTIC RESETTING

A system of three nonlinear equations that depict chaotic behavior was considered in this work. We looked at four cases: one with a power-law property; one with a fading memory effect; one with a transition from stretched exponential to power-law; and one with a stochastic component. We have presented a detailed analysis of stability and we have suggested a number similar to the reproductive number in the case of epidemiology. At this stage, we did not provide a physical meaning for such a number. However, such a number may be useful. A numerical scheme based on the Lagrange polynomial interpolation was used to solve the respective models numerically. Numerical simulations were performed for different values of fractional orders and densities of randomness and piecewise ideas.

A Robust Fractional-Order Control Technique for Stable Performance of Multilevel Converter-Based Grid-Tied DG Units

In this article, we propose a sliding fractional-order (FO) control strategy to provide robustness feature for integrating a multilevel converter into the power grid under the system parameter uncertainties, nonlinear load alterations, and grid voltage sag. All dynamics achieved from the input capacitors and output inductances of the multilevel converter are utilized to design the proposed sliding fractional surfaces (SFSs) such that a robust operation of the grid-tied converter-based system is achieved. Then, an SFS-based Lyapunov function is built to assess the proposed controller stability. A thorough analysis of the Lyapunov function is carried out for distinguishing suitable boundaries of the controller coefficients when the error components of the system state variables are changed, accordingly. Using the derivative of Lyapunov function and SFS, a comprehensive evaluation is executed to detail the regulation procedure for the robustness and fractional PI controller coefficients. Finally, a closed-loop system using the converter current dynamics aims to discern the differences between different values of FO. Experimental tests using dSPACE-1202 along with MATLAB/Simulink simulations are employed to verify the effectiveness of the proposed control technique for providing proper harmonic compensation, enhanced power quality, unity power factor, voltage sag tolerance, and very low total harmonic distortion grid current.

Stability Analysis of Fractional PSQp Smoking Model and Application in Turkey

Smoking is one of the most common habits in the world. Due to the harmful substances contained in it cigarette disturb people’s health and causes their death. Every year, many people lose their health from smoking or by direct exposure to cigarettes. In this study, we conducted a new application and stability analysis related to a fractional model of cigarette use. This model consists of three compartments: potential smokers (P), smokers (S) and defi-nitely quitters of smoking (Qp). The fractional derivative is used in the sense of Caputo. In this article, we develop a fractional mathematical model to illustrate the consequences of smoking habit in Turkey by the Caputo fractional-order derivative. In addition to stability analysis, numerical solutions are obtained by using Euler method for different values of fractional order

VARIATIONAL ITERATION METHOD FOR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS - A UNIVERSAL APPROACH BY SUMUDU TRANSFORM

The important feature of this research paper is an extension to solve linear and nonlinear applications of fractional partial differential equations suggested by D. Ziane and M. H. Cherif for various values of α (1 < α ≤ 2). The contemplated graphs show that the behavior of exact and approximate solution for different values of fractional order α. The effectiveness and convenience of the method is tested with the help of two illustrative examples. The fractional derivative is described in the Liouville-Caputo sense.

CHAOTIC BEHAVIOR OF MODIFIED STRETCH–TWIST–FOLD FLOW UNDER FRACTAL-FRACTIONAL DERIVATIVES

The application of the recently proposed integral and differential operators known as the fractal-fractional derivatives and integrals has opened doors to ongoing research in different fields of science, engineering, and technology. These operators are a convolution of the fractal derivative with the generalized Mittag-Leffler function with Delta-Dirac property, the power law, and the exponential decay law with Delta-Dirac property. In this paper, we aim to extend the work in the literature by applying these operators to a modified stretch–twist–fold (STF) flow based on the STF flow related to the motion of particles in fluids that naturally occur in the dynamo theorem. We want to capture the dynamical behavior of the modified STF flow under these operators. We will present the numerical schemes that can be used to solve these nonlinear systems of differential equations. We will also consider numerical simulations for different values of fractional order and fractal dimension.

Modified Three-Point Fractional Formulas with Richardson Extrapolation

In this paper, we introduce new three-point fractional formulas which represent three generalizations for the well-known classical three-point formulas; central, forward and backward formulas. This has enabled us to study the function’s behavior according to different fractional-order values of α numerically. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability.

New approximate series solutions of conformable time–space fractional Fokker–Planck Equation via two efficacious techniques

This article presents a comparative analysis of the conformable time–space fractional Fokker–Planck equation (TSF-FP equation) by operating two efficacious techniques: Fractional homotopy analysis transform method (FHATM) and conformable homotopy transform method (CHTM). The efficaciousness and validity of the proposed techniques are represented in terms of three different examples of conformable TSF-FP equations by computing various error norms: L2 and absolute error. Convergence analysis of both techniques are studied theocratically and verified graphically as well. The findings demonstrate that for a given order of approximation CHTM requires less computational time in comparison to FHATM while FHATM is more general and converges faster than CHTM. The obtained information is confirmed graphically for different numerical values of fractional order.

Swarming optimization to analyze the fractional derivatives and perturbation factors for the novel singular model

The aim of this research is to present an investigation based on the fractional derivatives and perturbation factors for the novel singular system. This study also presents a novel design of the fractional perturbed singular system by using the conventional Lane-Emden form together with the features of fractional order values, singular points, perturbed terms and shape factors. An analysis based on the fractional order derivative and perturbation factors is provided using the novel singular form of the Lane-Emden system in two different ways with three different variations. The numerical representations based on the novel design of the fractional perturbed singular system are presented through the Meyer wavelet neural networks (MWNNs). The optimization is performed by using the hybrid efficiency of the global swarming particle swarm optimization (PSO) scheme along with the local interior-point algorithm (IPA). The modeling through the MWNN is signified through the novel fractional perturbed singular system through the mean square error along with the PSOIPA optimization. The exactness, verification, endorsement and excellence of the novel fractional perturbed singular system are authenticated through the comparison of the obtained and the true solutions. The reliability of the stochastic procedure is performed by using the statistical measures with a large domain of the dataset to analyze the fractional derivatives and perturbation factors for the novel singular system.

Fractional study of a novel hyper-chaotic model involving single non-linearity

The applications of hyperchaotic systems (HCSs) can be widely seen in diverse fields associated with engineering due to their complicated dynamics, randomness, and high delicacy and sensibility. In the present work, we aim to investigate a new hyper-chaotic system involving a single non-linearity under the fractional Caputo–Fabrizio (CF) derivative for the first time. In fact, there is no previous study using fractional derivatives in this system. A new mathematical system using a fractional-order operator will be designed with the novel operator. The Caputo–Fabrizio non-integer operator is aimed to be employed to capture complex nature. In order to solve the extracted dynamical system, a quadratic numerical scheme is applied. This study contains stability and convergence sections for the considered method. Moreover, numerical results of the problem under various values of fractional orders and different values of initial conditions (ICs) are provided to show the performance of the suggested scheme. Figures of solutions for each dependent variable can be observed.

Dynamics of Middle East Respiratory Syndrome Coronavirus (MERS-CoV) involving fractional derivative with Mittag-Leffler kernel

Since 2012, the Middle East has seen a steady rise in the Middle East Respiratory Syndrome Coronavirus (MERS-CoV). A fractional derivative of the non-singular Mittag-Leffler type is used in this research to conduct a mathematical analysis of the dynamics of MERS-CoV infection transmission. The dynamics of such a disease with an additional degree of freedom and non-singular behavior are discovered through the use of the aforementioned fractional operator, and this is one of the important components of our prepared paper. Using the concept of fixed point theory, the existence and uniqueness of solutions are demonstrated. The stability analysis is also tested with the help of the Ulam–Hyers approach, respectively. The numerical solution has been conducted by using the fractional Adams–Bashforth scheme. In the numerical simulation, all classes are demonstrated through the graphical presentation regarding the changing values of fractional-order at time t. The results at various fractional-order laying between (0,1] are drawn with the help of Matlab. We also provide a comparison of the proposed approach with that of the Caputo operator. The outcomes that were achieved illustrate that the considered scheme is highly methodical and very efficient compared to the Caputo fractional operator.

Analysis of time-fractional non-linear Kawahara Equations with power law kernel

In this article, we study time-fractional non-linear Kawahara and modified Kawahara equations with Caputos fractional derivative. A variety of integral transforms with Adomian decomposition are applied to obtain the general series solutions of the considered models. The efficiency of the proposed methods is confirmed by numerical examples under suitable initial conditions. From the numerical results, one can see that the attained series solutions converge to the exact solutions of the systems. The stability of the applied methods is investigated using the principle of Banach contraction and S-stable mapping. For the Kawahara Equation, it is observed that the amplitude of the system enhances as time increases for fixed values of fractional orders. Similarly, for stable temporal variables, when the fractional-order increases, the amplitude of the solitary wave solution also increases. Similarly, for the modified system, the wave amplitude is also enhanced with variations in time (t). It infers that when (α) increases, it significantly decreases the wave amplitude. It is also observed that uniform changes take place in the wave amplitude with time (t). However, non-uniform changes in wave amplitude occur for different values of α. The absolute error between the exact and obtained series solutions is presented. It is revealed that the absolute error in the systems reduces promptly when x increases at a comparatively small time t, whereas the increment in iterations decreases the error in the systems.

Fractional-order circuit design with hybrid controlled memristors and FPGA implementation

A hybrid memristor circuit consisting of a magnetic-controlled and a charge-controlled is designed. In order to construct a relatively complex dynamical system, an absolute value and a square root algorithm are embedded in the magnetic-controlled model specially. This novel five-dimensional system is first analyzed with integer order for the existence of chaotic phenomena, and then implemented the fractional-order memristor system by the Grunwald–Letnikov algorithm. Five fractional order values of qi(i=1,2,3,4,5) are taken as identical and different in the numerical simulation, respectively. The bifurcation diagram, 0-1 test, SALI algorithm, and complexity analysis reveal the existence of nonlinear dynamic characteristics to the hybrid fractional-order memristor system. Finally, the commensurate and non-commensurate fractional-order systems based on the frequency domain method are implemented by FPGA technology, their accuracy can reach to 14 decimal places. The results of numerical calculation and circuit simulation have the same phase trajectory, which can verify that the hybrid memristors have practical values in the future.

ANALYSIS OF HIDDEN ATTRACTORS OF NON-EQUILIBRIUM FRACTAL-FRACTIONAL CHAOTIC SYSTEM WITH ONE SIGNUM FUNCTION

In 2017, Atangana proposed more generalized operators depending on two parameters: one is fractional order (FO) and other is fractal dimension. The novel operators are defined with three different kernels. These operators produced excellent dynamics of the chaotic systems. In this paper, the Caputo fractal-fractional operator is used to explore a chaotic system which contains only one signum function. The existence theory is developed by using the fixed-point result of Leray–Schauder to prove that the considered chaotic system possesses at least one solution. The proposed chaotic system has a unique solution, according to Banach’s fixed-point theorem. We demonstrate that the suggested chaotic system is Ulam–Hyres (UH) stable under the novel operator of power law kernel by employing nonlinear functional analysis. The Adams–Bashforth technique is used to evaluate the numerical outcomes of the considered model. We show the complex structure of numerical solutions for different FO and fractal dimension values.

The generalized projective Riccati equations method for solving quadratic-cubic conformable time-fractional Klien-Fock-Gordon equation

In this article, the generalized projective Riccati equations method is proposed for solving time-fractional Klien-Fock-Gordon (KFG) equation. The proposed model is explored for quadratic and cubic nonlinearities. The conformal derivative is used for time-fractional derivative in KFG equation. The soliton solutions are constructed and illustrated graphically for some particular values of fractional order α,0<α<1. The graphical illustration includes 3D plots and contour plots for obtained solutions. It has been observed that the mutation from quadratic-state to cubic-state causes change in the physical interpretation of obtained solutions.

The epidemic COVID-19 model via Caputo–Fabrizio fractional operator

In this article, we analyze and identify the optimum values for a deeper sense of the mathematical model of the COVID-19 epidemic from the reservoir to humans by using a powerful fractional homotopy perturbation transform method with Caputo–Fabrizio fractional derivative. We receive simulations of this propagation under high parameters. Although the results show the efficacy of the theoretic framework considered for the governing structure. The obtained results also provide lighting on the dynamic behavior of the COVID-19 model. We gave a few numerical approximations to explain the efficiency of the proposed method for various values of fractional order, which correspond to the process. Finally, we graphically demonstrate the obtained outcome.

The solution of fractional-order system of KdV equations with exponential-decay kernel

This study uses efficient techniques to evaluate a non-linear system of Korteweg–de Vries (KdV) equations with fractional Caputo Fabrizio derivative, including the modified decomposition approach and the novel iterative transform method. The system of KdV equations and the modify scheme of KdV equations used as a model in non-linear physical processes emerging in biology, chemistry, physics and sciences are the non-linear fractional coupled systems explored in this present analysis. Approximate analytical outcomes are represented as a series with simple components, and some features revealed a proper dependency on the fractional-order derivatives’ values. An examination of convergence and uniqueness is performed. Three test cases for the analytic findings of the fractional-order KdV equations are supplied to help understand the analytical technique of both methods. Furthermore, the efficiency of the aforementioned operations, as well as the decrease in computations, allow for a larger application. It is also demonstrated that the present methodology’s conclusions are in close agreement with the precise answers. With few computations, the series result obtained using this approach has been proven to be accurate and dependable. For various fractional-order values, numerical simulations for derived solutions are described.

Generalized fractional strategy for recommender systems with chaotic ratings behavior

To provide useful and accurate recommendations, the role of recommender systems for e-commerce industry is to predict users' interest by approximating users' preferences and items characteristics. The increasing role of recommender systems in the e-commerce industry requires efficient recommendations to deal with the chaos in the users' ratings patterns. In this regard, different fractional gradient based adaptive algorithms are employed through matrix factorization to improve the efficiency of recommender systems in terms of recommendations speed and accuracy. In this paper, the generalized variant of fractional stochastic gradient descent termed as GFSGD is exploited for recommender systems. The proposed GFSGD improves the performance of the recommender systems with the capability to enhance the memory effect by capturing the ratings history more effectively. The proposed GFSGD assures fast convergence speed for higher fractional order values while, taking fractional order unity reduces the GFSGD to the standard SGD. The correctness of the GFSGD is confirmed through various evaluation metrics (RMSE, MAE, sMAPE, NSE), learning rates, fractional orders, and latent features. While the exactness of the proposed scheme is also verified through Movie-Lens and FilmTrust datasets.